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NOTE: the formal title "Dominion", preserved in the names of the countries of Canada and New Zealand, does not refer to any supposed British dominance. Although it dates back to the foundation of those countries within the British Empire, the title is in fact of Canadian origin.

At the time of Confederation in 1867, the founders of modern Canada wanted to include in the formal title of their new country "a tribute to the monarchical principle, which they earnestly desire to uphold". Consequently they wanted to christen their country the "Kingdom of Canada". This was also expected to antagonise the Americans.

Not wanting to worsen diplomatic relations with the United States, the Imperial government in London asked the Canadians to choose a different form of words. They responded to London with "Dominion of Canada", a clever pun on Psalm 72 as both a description of Canada and kingdom: "He shall have dominion also from sea to sea, and from the river unto the ends of the earth". This was accepted.

Once established, the Dominion of Canada became the archetype for independent communities within the Empire, eventually fully-independent countries as the Empire changed into a Commonwealth. "Dominion" became the shorthand description for such countries, and remains in the formal titles of both Canada and New Zealand.

It does not imply, nor ever has been intended to imply,  hegemony from London. Indeed the Crown with respect to the Dominions rapidly and logically evolved into the modern system of independent Crowns for each country: the Canadian, Australian and New Zealand Crowns (for example) are independent at law from their British counterpart, although linked by the same monarch, currently Elizabeth II.

Some countries emerging from the British Empire, such as Australia and the Bahamas, preferred the ancient English political title of "Commonwealth". Made famous by Thomas Hobbes' political treatise Leviathan and the 17th Century English republic ("The Commonwealth of England"), this word refers to the idea of citizens forming a community, each surrendering some of his or her personal sovereignty to create a government with sovereignty over them all. (These different concepts of sovereignty are not incompatible: the modern Westminster Crown has been explicitly constructed by splicing Tory and republican concepts since the 17th Century, to answer weaknesses identified during and after the republican period, as well as those of the Stuart autocracies.) 

Because of lack of understanding of the history of the term "Dominion", the preferred generic description of such countries is now "Her Majesty's realms". 
         
Under the Statute of Westminster (1931) and related international conventions, the Parliament of each independent country under Elizabeth II stands co-equal, and the consent of each is required to modify relevant international legislation pertaining to the laws of succession of the shared monarch. The Queen's constitutional duties with respect to her other countries, states and provinces outside the UK are generally performed by her relevant governors-general, governors and lieutenant-governors, who are by convention often eminent members of the local community, commonly appointed by the Queen on the advice of that state's prime minister or premier (but sometimes elected).

In a nutshell, the Queen's job, and that of her representatives, is to be the ultimate executive guardian of parliamentary democracy, under conditions where other safeguards fail.

Consequently it's important that they not be politically biased (which is why electing them is generally a bad idea).There are some conventions that attempt to avoid politicisation of appointed governors-general; there are also other, more interesting ways available for choosing a viceregal representative, to ensure this more effectively: see For the Sovereignty of the People for a discussion.

The British government has no authority to attempt to advise the Queen with respect to her other realms and territories; furthermore, her representatives' relationship with the relevant democratically elected government is generally  analogous to the Queen's with her British ministers, although with some differences: again, see my book for a lengthy discussion of the relevant legal conventions.

Consequently her other Crowns, representatives and governments operate mutually independently, uninfluenced and unaffected by the UK or its government, in some of the world's most successful parliamentary democracies.